sorting uc berkeley fall 2004, e77 pack/e77 copyright 2005, andy packard. this work is licensed...

SortingUC Berkeley

Fall 2004, E77http://jagger.me.berkeley.edu/~pack/e77

Copyright 2005, Andy Packard. This work is licensed under the Creative Commons Attribution-ShareAlike License. To view a copy of this license, visit http://creativecommons.org/licenses/by-sa/2.0/ or send a letter to

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http://jagger.me.berkeley.edu/~pack/e77

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Sorting

Keeping data in “order” allows it to be searched more efficiently

Example: Phone Book–Sorted by Last Name (“lots” of work to do this)

• Easy to look someone up if you know their last name

• Tedious (but straightforward) to find by First name or Address

Important if data will be searched many times

Two algorithms for sorting today–BubbleSort–MergeSort

Searching: next lecture

Bubble Sort (“Sink” sort here)

If A(1)>A(2) switchIf A(2)>A(3) switchIf A(3)>A(4) switchIf A(4)>A(5) switch …If A(N-3)>A(N-2) switchIf A(N-2)>A(N-1) switchIf A(N-1)>A(N) switch

If A(1)>A(2) switchIf A(2)>A(3) switchIf A(3)>A(4) switchIf A(4)>A(5) switch

If A(N-3)>A(N-2) switchIf A(N-2)>A(N-1) switch


If A(N-3)>A(N-2) switch

If A(1)>A(2) switch

A(N) is now largest entry

A(N-1) is now 2nd largest entry

A(N) is still largest enry

A(N-2) is now 3rd largest entry

A(N-1) is still 2nd largest entry

A(N) is still largest enry

A(1) is now Nth largest entry.

A(2) is still (N-1)th largest entry.

A(3) is still (N-2)th largest entry.

A(N-3) is still 4th largest entry

A(N-2) is still 3rd largest entry

A(N-1) is still 2nd largest entry

A(N) is still largest entry







If A(1)>A(2) switch

N-1 steps

N-2 steps

N-3 steps

1 step

22

)1( steps of #

21

1

NNNi

N

i







If A(1)>A(2) switch

for lastcompare=N-1:-1:1

for i=1:lastcompare

if A(i)>A(i+1)

Matlab code for Bubble Sort

function S = bubblesort(A)

% Assume A row/column; Copy A to S

S = A;

N = length(S);


for i=1:lastcompare

if S(i)>S(i+1)

tmp = S(i);

S(i) = S(i+1);

S(i+1) = tmp;

end

end

end

What about returning an Index vector Idx, with the property that S = A(Idx)?

Matlab code for Bubble Sort

function [S,Idx] = bubblesort(A)

% Assume A row/column; Copy A to S

N = length(A);

S = A; Idx = 1:N; % A(Idx) equals S


for i=1:lastcompare

if S(i)>S(i+1)

tmp = S(i); tmpi = Idx(i);

S(i) = S(i+1); Idx(i) = Idx(i+1);

S(i+1) = tmp; Idx(i+1) = tmpi;

end

end

end

If we switch two entries of S, then exchange the same

two entries of Idx. This keeps A(Idx) equaling S

Merging two already sorted arrays

Suppose A and B are two sorted arrays (different lengths)

How do you “merge” these into a sorted array C?

Chalkboard…

Pseudo-code: Merging two already sorted arrays

function C = merge(A,B)

nA = length(A); nB = length(B);

iA = 1; iB = 1; %smallest unused element

C = zeros(1,nA+nB);

for iC=1:nA+nB

if A(iA)<B(iB) %compare smallest unused

C(iC) = A(iA); iA = iA+1; %use A

else

C(iC) = B(iB); iB = iB+1; %use B

end

end BA nn steps"" of #

MergeSort

function S = mergeSort(A)

n = length(A);

if n==1

S = A;

else

hn = floor(n/2);

S1 = mergeSort(A(1:hn));

S2 = mergeSort(A(hn+1:end));

S = merge(S1,S2);

end

Base Case

Split in half

Sort 2nd half

Merge 2 sorted arrays

Sort 1st half

Rough Operation Count for MergeSort

Let R(n) denote the number of operations necessary to sort (using mergeSort) an array of length n.

function S = mergeSort(A)

n = length(A);

if n==1

S = A;

else

hn = floor(n/2);

S1 = mergeSort(A(1:hn));

S2 = mergeSort(A(hn+1:end));

S = merge(S1,S2);

end

R(1) = 0 R(n/2) to sort array of length n/2

n steps to merge two sorted arrays of total length n

R(n/2) to sort array of length n/2

Recursive relation: R(1)=0, R(n) = 2*R(n/2) + n

Rough Operation Count for MergeSort

The recursive relation for R

R(1)=0, R(n) = 2*R(n/2) + n

Claim: For n=2m, it is true that R(n) ≤ n log2(n)

Case (m=0): true, since log2(1)=0

Case (m=k+1 from m=k)

12 1 kk

kkk 222log22 2 kkR 2222

kk RR 222 1

12

1 2log2 kk

Recursive relation

Induction hypothesis

Matlab command: sortSyntax is

[S] = sort(A)

If A is a vector, then S is a vector in ascending order

The indices which rearrange A into S are also available.

[S,Idx] = sort(A)

S is the sorted values of A, and A(Idx) equals S.

sorting uc berkeley fall 2004, e77 pack/e77 copyright 2005, andy packard. this work is licensed...

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